Relativistic Effects Non-collinear Magnetism (WIEN2k / WIENncm)
20th WIEN2k Workshop PennStateUniversity – 2013 Relativistic effects & Non-collinear magnetism (WIEN2k / WIENncm) Xavier Rocquefelte Institut des Matériaux Jean‐Rouxel (UMR 6502) Université de Nantes, FRANCE
20th WIEN2k Workshop PennStateUniversity – 2013 Talk constructed using the following documents: Slides of: Robert Laskowski, Stefaan Cottenier, Peter Blaha and Georg Madsen Notes of: - Pavel Novak (Calculation of spin-orbit coupling) http://www.wien2k.at/reg user/textbooks/ - Robert Laskowski (Non-collinear magnetic version of WIEN2k package) Books: - WIEN2k userguide, ISBN 3-9501031-1-2 - Electronic Structure: Basic Theory and Practical Methods, Richard M. Martin ISBN 0 521 78285 6 - Relativistic Electronic Structure Theory. Part 1. Fundamentals, Peter Schewerdtfeger, ISBN 0 444 51249 7 web: - y.html - wienlist digest - http://www.wien2k.at/reg user/index.html - wikipedia
Few words about Special Theory of Relativity Light Composed of photons (no mass) Speed of light constant Atomic units: ħ me e 1 c 137 au
Few words about Special Theory of Relativity Light Composed of photons (no Matter mass) Composed of atoms (mass) Speed of light constant Atomic units: ħ me e 1 c 137 au v f(mass) Speed of matter mass mass f(v)
Few words about Special Theory of Relativity Light Matter Composed of photons (no mass) Composed of atoms (mass) Speed of light constant Atomic units: ħ me e 1 v f(mass) Speed of matter mass mass f(v) c 137 au Lorentz Factor (measure of the relativistic effects) 1 v 1 c 2 1 Relativistic mass: M m (m: rest mass) Momentum: p mv Mv Total energy: E2 p2c2 m2c4 E mc2 Mc2
Definition of a relativistic particle (Bohr model) Lorentz factor ( ) Speed of the 1s electron (Bohr model): 10 9 « Non-relativistic » particle: 1 8 7 Ze e- 6 5 H(1s) 4 Au(1s) 3 Z ve n 2 1 0 20 40 60 80 Speed (v) Details for Au atom: ve (1s ) 79 c 0.58c 137 au HH: :vve e(1(1ss)) 11au 11.00003 .00003 Au : ve (1s ) 79 au Au : ve (1s ) 79 au 1.22 1.22 100 120 c 137 au 1 ve 1 c 2 1 1 0.58 1s electron of Au atom relativistic particle 2 1.22 Me(1s-Au) 1.22me
Relativistic effects Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c)
Relativistic effects Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung*) *Analysis of Erwin Schrödinger of the wave packet solutions of the Dirac equation for relativistic electrons in free space:The interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median.
Relativistic effects Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung) 3) The spin-orbit coupling It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l)
Relativistic effects Zeffe 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung) 3) The spin-orbit coupling It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l) 4) Indirect relativistic effect The change of the electrostatic potential induced by relativity is an indirect effect of the core electrons on the valence electrons
One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS 1 H S 2 V 2 2 2 H S V 2me Atomic units: ħ me e 1 1/(4 0) 1 c 1/ 137 au
One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS 1 H S 2 V 2 2 2 H S V 2me Z V r In a spherically symmetric potential Ze 2 V 4 0 r n ,l ,m Rn ,l r Yl ,m , 1 2 1 2 r 2 r r r r sin 2 Atomic units: ħ me e 1 1/(4 0) 1 c 1/ 137 au 2 1 sin r 2 sin 2 2
One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS 1 H S 2 V 2 2 2 V H S 2me Z V r In a spherically symmetric potential Ze 2 V 4 0 r n ,l ,m Rn ,l r Yl ,m , 1 2 1 2 r 2 r r r r sin 2 1 d 2 dRn ,l l l 1 r V 2 Rn ,l Rn ,l 2r dr dr 2r 2 2 1 sin r 2 sin 2 2 2 1 d 2 dRn ,l 2 l l 1 r V Rn ,l Rn ,l 2me r 2 dr 2me r 2 dr
Dirac Hamiltonian: a brief description Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. E2 p2c2 m2c4 H D with H D c p me c 2 V
Dirac Hamiltonian: a brief description Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. E2 p2c2 Momentum operator m2c4 H D with 0 k k 0 1 1 1 0 Rest mass H D c p me c 2 V k 0 1 0 k 0 1 0 i 2 i 0 (2 2) unit and zero matrices 1 0 3 0 1 (2 2) Pauli spin matrices Electrostatic potential
Dirac equation: HD and are 4-dimensional is a four-component single-particle wave function that describes spin-1/2 particles. In case of electrons: spin up spin down 1 1 Large 2 2 components ( ) HD 3 3 Small components ( ) 4 4 factor 1/(mec2) and are time-independent two-component spinors describing the spatial and spin-1/2 degrees of freedom Leads to a set of coupled equations for and : c p V me c 2 c p V me c 2
Dirac equation: HD and are 4-dimensional For a free particle (i.e. V 0): Solution in the slow particle limit (p 0) me c 2 0 pˆ z pˆ x ipˆ y 1 2 ˆ ˆ ˆ 0 me c p z ip y pz 2 0 pˆ 2 ˆ ˆ 0 p i p m c z z y e 3 2 pˆ ipˆ 0 me c 4 pˆ z z y Particles: up & down Non-relativistic limit decouples 1 from 2 and 3 from 4 0 me c 2 , 0 0 0 me c 2 , 0 0 Antiparticles: up & down 0 0 me c 2 , 0 0 0 me c 2 , 0
Dirac equation: HD and are 4-dimensional For a free particle (i.e. V 0): Solution in the slow particle limit (p 0) me c 2 0 pˆ z pˆ x ipˆ y 1 2 ˆ ˆ ˆ 0 me c p z ip y pz 2 0 pˆ 2 ˆ ˆ 0 p i p m c z z y e 3 2 pˆ ipˆ 0 me c 4 pˆ z z y Particles: up & down Non-relativistic limit decouples 1 from 2 and 3 from 4 0 me c 2 , 0 0 0 me c 2 , 0 0 Antiparticles: up & down 0 0 me c 2 , 0 0 0 me c 2 , 0 For a spherical potential V(r): g n r i f n r j l s 2 gn and fn are Radial functions Y are angular-spin functions s j 1 2 s 1, 1
Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn ) are simplified if we define: Energy: ' me c 2 Radially varying mass: M e r me ' V r 2c 2
Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn ) are simplified if we define: Energy: ' me c 2 Radially varying mass: M e r me ' V r 2c 2 Then the coupled equations can be written in the form of the radial eq.: 2 1 d 2 dg n r 2 2M e r dr dr 2 l l 1 2 dV dg n 2 dV 1 g n ' g n V g n 2 2 2 2 2 2M e r 4 M e c dr dr 4M e c dr r Mass-velocity effect 2 1 d 2 dRn ,l 2 l l 1 r V Rn ,l Rn ,l 2me r 2 dr dr 2me r 2 Note that: 1 l l 1 Darwin term Spin-orbit coupling One electron radial Schrödinger equation in a spherical potential
Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn ) are simplified if we define: Energy: ' me c 2 Radially varying mass: M e r me ' V r 2c 2 Then the coupled equations can be written in the form of the radial eq.: 2 1 d 2 dg n r 2 2M e r dr dr 2 l l 1 2 dV dg n 2 dV 1 g n ' g n V g n 2 2 2 2 2 2M e r 4 M e c dr dr 4M e c dr r and 1 f df nk 1 V ' g n n dr c r Darwin term Spin-orbit coupling Due to spin-orbit coupling, is not an eigenfunction of spin (s) and angular orbital moment (l). No approximation have been made Instead the good quantum numbers are j and so far Note that: 1 l l 1
Dirac equation in a spherical potential Scalar relativistic approximation Approximation that the spin-orbit term is small neglect SOC in radial functions (and treat it by perturbation theory) f f No SOC Approximate radial functions: g g n nl n nl 2 1 d 2 dg nl 2 l l 1 2 dV dg nl ' g nl r V g nl 2 2 2 2 dr 2M e r dr 2M e r 4M e c dr dr and f nl dg nl with the normalization condition: 2 M e c dr g 2 nl f nl2 r 2 dr 1
Dirac equation in a spherical potential Scalar relativistic approximation Approximation that the spin-orbit term is small neglect SOC in radial functions (and treat it by perturbation theory) f f No SOC Approximate radial functions: g g n nl n nl 2 1 d 2 dg nl 2 l l 1 2 dV dg nl ' g nl r V g nl 2 2 2 2 dr 2M e r dr 2M e r 4M e c dr dr and f nl dg nl with the normalization condition: 2 M e c dr g 2 nl f nl2 r 2 dr 1 The four-component wave function is now written as: Inclusion of the spin-orbit coupling in “second r g nl lm variation” (on the large component only) i f nl r lm H H SO with is a pure spin state 2 1 dV l 0 H is a mixture of up and down spin states SO 2 4M e c 2 r dr 0 0
Relativistic effects in a solid For a molecule or a solid: Relativistic effects originate deep inside the core. It is then sufficient to solve the relativistic equations in a spherical atomic geometry (inside the atomic spheres of WIEN2k). Justify an implementation of the relativistic effects only inside the muffin-tin atomic spheres
Implementation in WIEN2k Atomic sphere (RMT) Region Core electrons Valence electrons « Fully » relativistic Scalar relativistic (no SOC) Spin-compensated Dirac equation Possibility to add SOC (2nd variational) SOC: Spin orbit coupling
Implementation in WIEN2k Atomic sphere (RMT) Region Interstitial Region Core electrons Valence electrons Valence electrons « Fully » relativistic Scalar relativistic (no SOC) Not relativistic Spin-compensated Dirac equation Possibility to add SOC (2nd variational) SOC: Spin orbit coupling
Implementation in WIEN2k: core electrons Core states: fully occupied spin-compensated Dirac equation (include SOC) Atomic sphere sphere (RMT) (RMT) Region Region Atomic Core Core electrons electrons For spin-polarized potential, spin up and spin down are calculated separately, the density is averaged according to the occupation number specified in case.inc file. « Fully » relativistic Spin-compensated Dirac equation j l s/2 l -s(j 1/2) occupation s -1 s 1 s -1 s 1 s -1 s 1 s 0 1/2 -1 2 p 1 1/2 3/2 1 -2 2 4 d 2 3/2 5/2 2 -3 4 6 f 3 5/2 7/2 3 -4 6 8 case.inc for Au atom 17 0.00 0 1,-1,2 ( n, ,occup) 2,-1,2 ( n, ,occup) 2, 1,2 ( n, ,occup) 2,-2,4 ( n, ,occup) 3,-1,2 ( n, ,occup) 3, 1,2 ( n, ,occup) 3,-2,4 ( n, ,occup) 3, 2,4 ( n, ,occup) 3,-3,6 ( n, ,occup) 4,-1,2 ( n, ,occup) 4, 1,2 ( n, ,occup) 4,-2,4 ( n, ,occup) 4, 2,4 ( n, ,occup) 4,-3,6 ( n, ,occup) 5,-1,2 ( n, ,occup) 4, 3,6 ( n, ,occup) 4,-4,8 ( n, ,occup) 0
Implementation in WIEN2k: core electrons Core states: fully occupied spin-compensated Dirac equation (include SOC) case.inc for Au atom 1s1/2 2s1/2 Atomic sphere sphere (RMT) (RMT) Region Region Atomic Core Core electrons electrons For spin-polarized potential, spin up and spin down are calculated separately, the density is averaged according to the occupation number specified in case.inc file. « Fully » relativistic Spin-compensated Dirac equation j l s/2 l -s(j 1/2) occupation s -1 s 1 s -1 s 1 s -1 s 1 s 0 1/2 -1 2 p 1 1/2 3/2 1 -2 2 4 d 2 3/2 5/2 2 -3 4 6 f 3 5/2 7/2 3 -4 6 8 2p1/2 2p3/2 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 4s1/2 4p1/2 4p3/2 4d3/2 4d5/2 5s1/2 4f5/2 4f7/2 17 0.00 0 1,-1,2 ( n, ,occup) 2,-1,2 ( n, ,occup) 2, 1,2 ( n, ,occup) 2,-2,4 ( n, ,occup) 3,-1,2 ( n, ,occup) 3, 1,2 ( n, ,occup) 3,-2,4 ( n, ,occup) 3, 2,4 ( n, ,occup) 3,-3,6 ( n, ,occup) 4,-1,2 ( n, ,occup) 4, 1,2 ( n, ,occup) 4,-2,4 ( n, ,occup) 4, 2,4 ( n, ,occup) 4,-3,6 ( n, ,occup) 5,-1,2 ( n, ,occup) 4, 3,6 ( n, ,occup) 4,-4,8 ( n, ,occup) 0
Implementation in WIEN2k: valence electrons Valence electrons INSIDE atomic spheres are treated within scalar relativistic approximation [1] if RELA is specified in case.struct file (by default). Title F LATTICE,NONEQUIV.ATOMS: MODE OF CALC RELA unit bohr Atomic sphere sphere (RMT) (RMT) Region Region Atomic 1 225 Fm-3m 7.670000 7.670000 7.670000 90.000000 90.000000 90.000000 ATOM 1: X 0.00000000 Y 0.00000000 Z 0.00000000 MULT 1 ISPLIT 2 Au1 NPT 781 R0 0.00000500 RMT 2.6000 Z: 79.0 LOCAL ROT MATRIX: 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000 48 NUMBER OF SYMMETRY OPERATIONS Valence Valence electrons electrons Scalar relativistic (no SOC) no dependency of the wave function, (n,l,s) are still good quantum numbers all relativistic effects are included except SOC small component enters normalization and calculation of charge inside spheres augmentation with large component only SOC can be included in « second variation » [1] Koelling and Harmon, J. Phys. C (1977) Valence electrons in interstitial region are treated classically
Implementation in WIEN2k: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw1): H1 1 1 1 - Second diagonalization (lapwso): H1 H SO The second equation is expanded in the basis of first eigenvectors ( 1) N j ij 1 1j H SO 1i i 1 1j i sum include both up/down spin states N is much smaller than the basis size in lapw1 Atomic sphere sphere (RMT) (RMT) Region Region Atomic Valence Valence electrons electrons Scalar relativistic (no SOC) Possibility to add SOC (2nd variational)
Implementation in WIEN2k: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw1): H1 1 1 1 - Second diagonalization (lapwso): H1 H SO The second equation is expanded in the basis of first eigenvectors ( 1) N j ij 1 1j H SO 1i i 1 Atomic sphere sphere (RMT) (RMT) Region Region Atomic 1j i sum include both up/down spin states N is much smaller than the basis size in lapw1 Valence Valence electrons electrons Scalar relativistic (no SOC) Possibility to add SOC (2nd variational) SOC is active only inside atomic spheres, only spherical potential (VMT) is taken into account, in the polarized case spin up and down parts are averaged. Eigenstates are not pure spin states, SOC mixes up and down spin states Off-diagonal term of the spin-density matrix is ignored. It means that in each SCF cycle the magnetization is projected on the chosen direction (from case.inso) VMT: Muffin-tin potential (spherically symmetric)
Controlling spin-orbit coupling in WIEN2k Do a regular scalar-relativistic “scf” calculation save lapw initso lapw case.inso: WFFIL 4 1 0 -10.0000 1.50000 0. 0. 1. NX NX1 -4.97 0.0005 0 0 0 0 0 llmax,ipr,kpot emin,emax (output energy window) direction of magnetization (lattice vectors) number of atoms for which RLO is added atom number,e-lo,de (case.in1), repeat NX times number of atoms for which SO is switch off; atoms case.in1(c): ( ) 2 0.30 0.005 CONT 1 0 0.30 0.000 CONT 1 K-VECTORS FROM UNIT:4 -9.0 4.5 65 emin/emax/nband symmetso (for spin-polarized calculations only) run(sp) lapw -so -so switch specifies that scf cycles will include SOC
Controlling spin-orbit coupling in WIEN2k The w2web interface is helping you Non-spin polarized case
Controlling spin-orbit coupling in WIEN2k The w2web interface is helping you Spin polarized case
Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-Be Z 4 Bulk modulus: - NREL: 131.4 GPa - SREL: 131.5 GPa - Exp.: 130 GPa
Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-Be Z 4 Bulk modulus: - NREL: 131.4 GPa - SREL: 131.5 GPa - Exp.: 130 GPa LDA overbinding (2%) Clear difference NREL/SREL hcp-Os Z 76 Bulk modulus: - NREL: 344 GPa - SREL: 447 GPa - Exp.: 462 GPa
Relativistic effects in the solid: Illustration hcp-Be Z 4 Scalar-relativistic (SREL): hcp-Os Z 76 - LDA overbinding (2%) - Bulk modulus: 447 GPa spin-orbit coupling (SREL SO): - LDA overbinding (1%) - Bulk modulus: 436 GPa Exp. Bulk modulus: 462 GPa
(1) Relativistic orbital contraction Radius of the 1s orbit (Bohr model): r2 (e/bohr) 50 Non relativistic (l 0) 40 e- 30 Au 1s Ze n 2 a0 1 bohr a r (1s) AND 0 me c Z 20 10 r (1s) 0 0.00 0.01 0.02 0.03 r (bohr) Atomic units: ħ me e 1 c 1/ 137 au 0.04 0.05 0.06 1 0.013 bohr 79
(1) Relativistic orbital contraction Radius of the 1s orbit (Bohr model): r2 (e/bohr) 50 Non relativistic (l 0) Relativistic ( -1) 40 e- 30 Ze Au 1s n 2 a0 1 bohr r (1s) AND a0 Z mc 20 10 20% Orbital contraction r (1s) 0 0.00 0.01 0.02 0.03 r (bohr) 0.04 0.05 0.06 1 0.013 bohr 79 In Au atom, the relativistic mass (M) of the 1s electron is 22% larger than the rest mass (m) n 2 a0 1 1 0.010 bohr r (1s ) Z 79 1.22 a0 RELA M me 1.22me a 0 M e c
(1) Relativistic orbital contraction r2 (e/bohr) 0.5 Non relativistic (l 0) Relativistic ( -1) 0.4 ve (6s) Z 79 13.17 0.096c n 6 0.3 Orbital contraction 0.2 0.1 Au 6s 1 v 1 e c 2 1 1 0.096 2 1.0046 0.0 0 2 4 r (bohr) 6 Direct relativistic effect (mass enhancement) contraction of 0.46% only However, the relativistic contraction of the 6s orbital is large ( 20%) ns orbitals (with n 1) contract due to orthogonality to 1s
(1) Orbital Contraction: Effect on the energy r2 (e/bohr) r2 (e/bohr) 0.5 50 Relativistic correction (%) ERELA ENRELA E NRELA 20 -20 -30 -40 30 Non relativistic (l 0) Relativistic ( -1) 0.4 0.3 Orbital contraction 20 Orbital contraction 0.2 0.1 Au 1s 0 Au 6s 0.0 0.00 1s 2s -10 40 10 10 0 Non relativistic (l 0) Relativistic ( -1) 0.01 3s 0.02 0.03 r (bohr) 0.04 4s 0.05 0.06 0 5s 2 4 r ((bohr)) 6 6s
(2) Spin-Orbit splitting of p states r2 (e/bohr) Non relativistic (l 1) 0.7 0.6 0.5 0.4 Au 5p 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 r (bohr) 2.0 2.5
(2) Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r2 (e/bohr) E Non relativistic (l 1) Relativistic ( -2) 0.7 j 3/2 ( -2) 0.6 l 1 0.5 0.4 Au 5p 0.3 j 1 1/2 3/2 0.2 orbital moment 0.1 0.0 0.0 0.5 1.0 1.5 r (bohr) 2.0 2.5 e spin -e p3/2 ( -2): nearly same behavior than non-relativistic p-state
(2) Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r2 (e/bohr) 0.7 Non relativistic (l 1) 0.6 Relativistic ( 1) E l 1 0.5 j 1/2 ( 1) 0.4 Au 5p 0.3 j 1-1/2 1/2 0.2 orbital moment 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 r (bohr) p1/2 ( 1): markedly different behavior than non-relativistic p-state g 1 is non-zero at nucleus e -e spin
(2) Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r2 (e/bohr) E Non relativistic (l 1) Relativistic ( -2) Relativistic ( 1) 0.7 0.6 j 3/2 ( -2) l 1 0.5 j 1/2 ( 1) 0.4 Au 5p 0.3 j 1 1/2 3/2 0.2 orbital moment 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 r (bohr) e j 1-1/2 1/2 spin orbital moment -e Ej 3/2 Ej 1/2 p1/2 ( 1): markedly different behavior than non-relativistic p-state g 1 is non-zero at nucleus e -e spin
(2) Spin-Orbit splitting of p states r2 (e/bohr) Non relativistic (l 1) Relativistic ( -2) Relativistic ( 1) 0.7 0.6 Relativistic correction (%) 0.5 ERELA ENRELA 0.4 E NRELA 0.3 Au 5p 0.2 20 0.1 10 0.0 2p1/2 0.0 2p3/2 1 -2 3p1/2 3p3/2 4p1/2 4p3/2 0.5 5p1/2 5p3/2 1.0 1.5 2.0 r (bohr) 0 -10 -20 -30 -40 Scalar-relativistic p-orbital is similar to p3/2 wave function, but does not contain p1/2 radial basis function 2.5
(3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) -e Ze -e -e
(3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) -e Ze -e -e Zeff1 Z- (NREL) Zeff1e -e
(3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) Relativistic (REL) -e -e Ze Ze -e -e Zeff1 Z- (NREL) -e Zeff1 Zeff2 Zeff1e -e Zeff2 Z- (REL) Zeff2e -e -e
(3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) Relativistic (REL) -e -e Ze Ze -e -e Zeff1 Z- (NREL) -e -e Zeff1 Zeff2 Zeff1e Zeff2 Z- (REL) Zeff2e -e Indirect relativistic effect -e
(3) Orbital expansion: Au(d) states Relativistic correction (%) 5d3/2 5d5/2 ERELA ENRELA 20 4f5/2 4f7/2 3 -4 E NRELA 3d3/2 3d5/2 2 -3 10 4d3/2 4d5/2 0 r2 (e/bohr) r2 (e/bohr) -10 4 -20 3 0.3 2 0.2 -30 -40 Non relativistic (l 2) Relativistic ( 2) Relativistic ( -3) 1 Non relativistic (l 2) Relativistic ( 2) Relativistic ( -3) 0.4 Orbital expansion 0.1 Au 3d Au 5d 0 0.0 0.0 0.1 0.2 r (bohr) 0.3 0.4 0 1 2 r (bohr) 3 4
Relativistic effects on the Au energy levels Relativistic correction (%) ERELA ENRELA E NRELA 5d3/2 5d5/2 4f5/2 4f7/2 20 3d3/2 3d5/2 10 2p1/2 2p3/2 3p1/2 3p3/2 1s 2s 0 -10 -20 -30 -40 3s 4d3/2 4d5/2 4p1/2 4p3/2 4s 5p1/2 5p3/2 5s 6s
Atomic spectra of gold SO splitting SO splitting
Ag – Au: the differences (DOS & optical prop.) Ag Au
Relativistic semicore states: p1/2 orbitals Electronic structure of fcc Th, SOC with 6p1/2 local orbital Energy vs. basis size DOS with and without p1/2 6p1/2 6p3/2 p1/2 not included 6p1/2 p1/2 included 6p3/2 J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 153102 (2001)
SOC in magnetic systems SOC couples magnetic moment to the lattice direction of the exchange field matters (input in case.inso) Symmetry operations acts in real and spin space number of symmetry operations may be reduced (reflections act differently on spins than on positions) time inversion is not symmetry operation (do not add an inversion for klist) initso lapw (must be executed) detects new symmetry setting Direction of magnetization [100] [010] [001] [110] 1 A A A A mx A B B - my B A B - 2z B B A B
Relativity in WIEN2k: Summary WIEN2k offers several levels of treating relativity: non-relativistic: select NREL in case.struct (not recommended) standard: fully-relativistic core, scalar-relativistic valence mass-velocity and Darwin s-shift, no spin-orbit interaction ”fully”-relativistic: adding SO in “second variation” (using previous eigenstates as basis) adding p1/2 LOs to increase accuracy (caution!!!) x lapw1 (increase E-max for more eigenvalues, to have x lapwso basis for lapwso) x lapw2 –so -c SO ALWAYS needs complex lapw2 version Non-magnetic systems: SO does NOT reduce symmetry. initso lapw just generates case.inso and case.in2c. Magnetic systems: symmetso dedects proper symmetry and rewrites case.struct/in*/clm*
CuO interlude ATOMIC STRUCTURE OF CuO Cu O CuO4 square planar
CuO interlude ATOMIC STRUCTURE OF CuO Cu O CuO2 ribbons
CuO interlude ATOMIC STRUCTURE OF CuO Cu O Oxygen 4-fold coordinated
CuO interlude ATOMIC STRUCTURE OF CuO Cu O Monoclinic 3D atomic structure
CuO interlude MAGNETIC STRUCTURE OF CuO Cu O
CuO interlude MAGNETIC STRUCTURE OF CuO a ‐ Cu ‐ O c ‐
CuO interlude LOW-TEMPERATURE MAGNETIC STRUCTURE OF CuO FROM SINGLE-CRYSTAL NEUTRON DIFFRACTION[1] c a Magnetic moments are along the [0 1 0] direction ‐ ‐ ‐ ‐ Cu Cu ‐ ‐ O ‐ [1] J.B. Forsyth et al., J. Phys. C: Solid State Phys. 21 (1988) 2917 ‐ AFM interactions along [1 0 -1] FM interactions along [1 0 1]
CuO interlude Estimation of the Magneto-crystalline Anisotropy Energy (MAE) of CuO Allows to define the magnetization easy and hard axes Here we have considered the following expression: MAE E[u v w] – E[0 1 0] E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw] crystallographic direction [1] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm., Accepted
20th WIEN2k Workshop PennStateUniversity – 2013 Relativistic effects & Non-collinear magnetism (WIEN2k / WIENncm) Xavier Rocquefelte Institut des Matériaux Jean‐Rouxel (UMR 6502) Université de Nantes, FRANCE
Pauli Hamiltonian for magnetic systems 2x2 matrix in spin space, due to Pauli spin operators 2 2 HP Veff B Beff l . 2me 0 1 1 1 0 0 i 2 i 0 1 0 3 0 1 (2 2) Pauli spin matrices
Pauli Hamiltonian for magnetic systems 2x2 matrix in spin space, due to Pauli spin operators 2 2 HP Veff B Beff l . 2me 0 1 1 1 0 0 i 2 i 0 1 0 3 0 1 (2 2) Pauli spin matrices Wave function is a 2-component vector (spinor) – It corresponds to the large components of the dirac wave function (small components are neglected) 1 1 H P 2 2 spin up spin down
Pauli Hamiltonian for magnetic systems 2x2 matrix in spin space, due to Pauli spin operators 2 2 HP Veff B Beff l . 2me Effective electrostatic potential Effective magnetic field Veff Vext VH Vxc Beff Bext Bxc Exchange-correlation potential Exchange-correlation field
Pauli Hamiltonian for magnetic systems 2x2 matrix in spin space, due to Pauli spin operators 2 2 HP Veff B Beff l . 2me Effective electrostatic potential Effective magnetic field Veff Vext VH Vxc Beff Bext Bxc Exchange-correlation potential Exchange-correlation field Many-body effects which are defined within DFT LDA or GGA Spin-orbit coupling 2 1 dV 2M e2 c 2 r dr
Exchange and correlation From DFT exchange correlation energy: hom E xc r , m r xc r , m dr 3 Local function of the electronic density ( ) and the magnetic moment (m) Definition of Vxc and Bxc (functional derivatives): E xc , m Vxc E xc , m Bxc m LDA expression for Vxc and Bxc: , m hom Vxc xc , m hom xc hom xc , m Bxc mˆ m Bxc is parallel to the magnetization density vector (m)
Non-collinear magnetism Direction of magnetization vary in space, thus spin-orbit term is present 2 2 HP Veff B Beff l . 2me 2 2 B Bx iB y Veff B Bz . 2me 2 2 B Bx iB y Veff B Bz . 2me 1 2 1 and 2 are non-zero Solutions are non-pure spinors Non-collinear magnetic moments
Collinear magnetism Magnetization in z-direction / spin-orbit is not present 2 2 HP Veff B Beff l . 2me 2 2 Veff B Bz . 0 2me 2 2 0 Veff B Bz . 2me 1 0 0 2 Solutions are pure spinors Collinear magnetic moments Non-degenerate energies
Non-magnetic calculation No magnetization present, Bx By Bz 0 and no spin-orbit coupling 2 2 HP Veff B Beff l . 2me 2 2 Veff 2me 0 0 0 0 2 2 Veff 2me Solutions are pure spinors Degenerate spin solutions
Magnetism and WIEN2k Wien2k can only handle collinear or non-magnetic cases non-magnetic case DOS EF magnetic case m n – n 0 m n – n 0 run lapw script: run lapw script: x x x x x x x x x x x x x lapw0 lapw1 lapw2 lcore mixer DOS lapw0 lapw1 –up lapw1 -dn la
Few words about Special Theory of Relativity Light Matter Composed of photons (no mass) Lorentz Factor (measure of the relativistic effects) Speed of light constant c . Definition of a relativistic particle (Bohr model) H:v (1s) 1au e Au:v (1s) 79au e 1.00003
9.2 Collinear Points 9.2.1 Collinear or Non-collinear Points in the Plane Two or more than two points which lie on the same straight line are called collinear points with respect to that line; otherwise they are called non-collinear. Let m be a line, then all the points on line m are collinear. In the given figure, the points P and Q are .
Geometry Midterm Review revised 12-9-15 _ 1. What is a net for the figure below? A. C. B. D. _ 2. What are the names of three collinear points? A. Points D, J, and B are collinear. C. Points A, J, and B are collinear. B. Points D, J, and K are collinear. D. Points L, J, and K are collinear. L J D A B K
quantum mechanics relativistic mechanics size small big Finally, is there a framework that applies to situations that are both fast and small? There is: it is called \relativistic quantum mechanics" and is closely related to \quantum eld theory". Ordinary non-relativistic quan-tum mechanics is a good approximation for relativistic quantum mechanics
Name three collinear points on line o 7. Name 4 sets of collinear points 8. Name two opposite rays on line j with endpoint R 9. Which 4 points are collinear? What line are they on? 10. How many points are there on line j? Points & Lines: CLASSWORK Name a point that is collinear with the given points 11. O and S 12. P and R 13. U and T 14. U and .
Introduction to Relativistic Quantum Fields Jan Smit Institute for Theoretical Physics University of Amsterdam Valckenierstraat 65, 1018XE Amsterdam . J.D. Bjorken and S.D. Drell, I: Relativistic Quantum Mechanics, McGraw-Hill (1964). J.D. Bjorken and S.D. Drell, II: Relativistic Quantum Fields, McGraw-Hill
Magnetism (Section 5.12) The subjects of magnetism and electricity developed almost independently of each other until 1820, when a Danish physicist named Hans Christian Oersted discovered in a classroom demonstration that an electric current affects a magnetic compass. He saw that magnetism was related to electricity.
diagrams starting from relativistic wave equations. This approach lacks math-ematical rigor, but is more intuitive. We start by reviewing the most important principles of non-relativistic quantum mechanics. 2 Schr odinger equation The wave equation
IEE Colloquia: Electromagnetic Compatibility for Automotive Electronics 28 September 1999 6 Conclusions This paper has briefly described the automotive EMC test methods normally used for electronic modules. It has also compared the immunity & emissions test methods and shown that the techniques used may give different results when testing an identical module. It should also be noted that all .
